In this paper, we take into account the black-hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. At first, we consider the cosmological constant as a dynamical pressure to study the phase transitions and analogy of the black holes with the Van der Waals liquid–gas system in the extended phase space. We make a comparison between linear and nonlinear electrodynamics and show that the lowest critical temperature belongs to Maxwell theory. Also, we make some arguments regarding how power of nonlinearity brings the system to Schwarzschild-like and Reissner–Nordström-like limitations. Next, we study the critical behavior of the system in the context of heat capacity. We show that critical behavior of system is similar to the one in phase diagrams of extended phase space. We also extend the study of phase transition points through geometrical thermodynamics (GTs). We introduce two new thermodynamical metrics for extended phase space and show that divergencies of thermodynamical Ricci scalar (TRS) of the new metrics coincide with phase transition points of the system. Then, we introduce a new method for obtaining critical pressure and horizon radius by considering denominator of the heat capacity.
Generalized Einstein-Maxwell theory: Seeley-DeWitt coefficients and logarithmic corrections to the entropy of extremal and nonextremal black holes
